[Paper Review] Intrinsic mirror symmetry and categorical crepant resolutions

Daniel Pomerleano|arXiv (Cornell University)|Mar 1, 2021

Geometric and Algebraic Topology58 references4 citations

TL;DR

This paper establishes finiteness properties of symplectic and Floer cohomology in affine log Calabi-Yau varieties, proving that degree-zero symplectic cohomology $SH^0(X)$ is finitely generated and acts as a filtered deformation of a combinatorially defined Stanley-Reisner ring. It further shows that wrapped Floer groups are finitely generated modules over $SH^0(X)$, providing a symplectic counterpart to algebraic finiteness conditions in mirror symmetry and enabling a categorical crepant resolution of $\operatorname{Spec}(SH^0(X))$ when $X$ is maximally degenerate with a homological section.

ABSTRACT

The main result of the present paper concerns finiteness properties of Floer theoretic invariants on affine log Calabi-Yau varieties $X$. Namely, we show that: (a) the degree zero symplectic cohomology $SH^0(X)$ is finitely generated and is a filtered deformation of a certain algebra defined combinatorially in terms of a compactifying divisor $\mathbf{D}.$ (b) For any Lagrangian branes $L_0, L_1$, the wrapped Floer groups $WF^*(L_0,L_1)$ are finitely generated modules over $SH^0(X).$ We then describe applications of this result to mirror symmetry, the first of which is an ``automatic generation" criterion for the wrapped Fukaya category $\mathcal{W}(X)$. We also show that, in the case where $X$ is maximally degenerate and admits a ``homological section", $\mathcal{W}(X)$ gives a categorical crepant resolution of the potentially singular variety $\operatorname{Spec}(SH^0(X))$. This provides a link between the intrinsic mirror symmetry program of Gross and Siebert and the categorical birational geometry program initiated by Bondal-Orlov and Kuznetsov.

Motivation & Objective

• To establish finiteness properties of Floer-theoretic invariants on affine log Calabi-Yau varieties, particularly $SH^0(X)$ and $WF^*(L_0,L_1)$.
• To connect symplectic invariants on $X$ to algebraic finiteness conditions expected in the mirror symmetry correspondence.
• To demonstrate that the wrapped Fukaya category $\mathcal{W}(X)$ provides a categorical crepant resolution of $\operatorname{Spec}(SH^0(X))$ under maximally degenerate and homological section conditions.
• To bridge the intrinsic mirror symmetry program of Gross-Siebert with the categorical birational geometry program of Bondal-Orlov and Kuznetsov.

Proposed method

• Use of symplectic cohomology $SH^*(X)$ as a Hamiltonian Floer cohomology invariant for exact, convex symplectic manifolds, equipped with a pair-of-pants product making it a unital ring.
• Application of wrapped Floer cohomology $WF^*(L_0,L_1)$ as modules over $SH^*(X)$ for Lagrangian branes $L_0, L_1$ in $X$.
• Construction of a combinatorial algebra $\mathcal{A}_\mathbf{k}$ via direct sum over strata of the compactifying divisor $\mathbf{D}$, equipped with a product structure via intersection maps.
• Definition of a filtered deformation of $\mathcal{A}_\mathbf{k}$, with the associated graded ring being the Stanley-Reisner ring of the divisor configuration.
• Use of $\mathbf{G}_m$-equivariance and filtration constraints from NEF divisors to restrict the form of the deformation, leading to a finite-dimensional moduli space of mirror families.
• Proof of isomorphism between the deformed ring and the symplectic cohomology $SH^0(X)$ via surjectivity and vanishing of the kernel in the associated graded.

Experimental results

Research questions

• RQ1 Is the degree-zero symplectic cohomology $SH^0(X)$ of an affine log Calabi-Yau variety finitely generated over a field $\mathbf{k}$?
• RQ2 Are the wrapped Floer groups $WF^*(L_0,L_1)$ finitely generated modules over $SH^0(X)$?
• RQ3 Can the wrapped Fukaya category $\mathcal{W}(X)$ serve as a categorical crepant resolution of $\operatorname{Spec}(SH^0(X))$ under maximally degenerate and homological section conditions?
• RQ4 How do the finiteness properties of symplectic cohomology mirror the algebraic finiteness conditions in the mirror algebraic variety $X^\vee$?

Key findings

• The degree-zero symplectic cohomology $SH^0(X,\mathbf{k})$ is a finitely generated $\mathbf{k}$-algebra.
• The full symplectic cohomology $SH^*(X,\mathbf{k})$ is a finitely generated module over $SH^0(X,\mathbf{k})$.
• For any pair of Lagrangian branes $L_0, L_1$, the wrapped Floer group $WF^*(L_0,L_1)$ is a finitely generated module over $SH^0(X)$.
• When $X$ is maximally degenerate and admits a homological section, $\mathcal{W}(X)$ gives a categorical crepant resolution of $\operatorname{Spec}(SH^0(X))$.
• The mirror family of the example with three divisors $D_1,D_2,D_3$ is isomorphic to a hypersurface defined by $u(x_1x_2x_3 - u) = 0$, with no smooth members due to an $\mathbb{A}^1$-worth of singularities.
• The deformation of the Stanley-Reisner ring is constrained to a 7-parameter family $g_{\vec{a}}$, and the resulting ring is isomorphic to $\mathcal{A}_\Lambda$ via a filtered isomorphism.